402 Maximum Likelihood Methods
(a)Show that the likelihood ratio for testingH 0 :θ 1 =θ 2 ,θ 3 =θ 4 against all
alternatives is given by
[n
∑
1
(xi−x)^2 /n
]n/ 2 [m
∑
1
(yi−y)^2 /m
]m/ 2
{[n
∑
1
(xi−u)^2 +
∑m
1
(yi−u)^2
]/
(m+n)
}(n+m)/ 2 ,
whereu=(nx+my)/(n+m).
(b)Show that the likelihood ratio test for testingH 0 :θ 3 =θ 4 ,θ 1 andθ 2 unspec-
ified, againstH 1 :θ 3
=θ 4 ,θ 1 andθ 2 unspecified, can be based on the random
variable
F=
∑n
1
(Xi−X)^2 /(n−1)
∑m
1
(Yi−Y)^2 /(m−1)
.
6.5.6.LetX 1 ,X 2 ,...,XnandY 1 ,Y 2 ,...,Ymbe independent random samples from
the two normal distributionsN(0,θ 1 )andN(0,θ 2 ).
(a)Find the likelihood ratio Λ for testing the composite hypothesisH 0 :θ 1 =θ 2
against the composite alternativeH 1 :θ 1
=θ 2.
(b)This Λ is a function of whatF-statistic that would actually be used in this
test?
6.5.7.LetXandYbe two independent random variables with respective pdfs
f(x;θi)=
{(
1
θi
)
e−x/θi 0 <x<∞, 0 <θi<∞
0elsewhere,
fori=1,2. To testH 0 :θ 1 =θ 2 againstH 1 :θ 1
=θ 2 , two independent samples
of sizesn 1 andn 2 , respectively, were taken from these distributions. Find the
likelihood ratio Λ and show that Λ can be written as a function of a statistic having
anF-distribution, underH 0.
6.5.8. For a numerical example of theF-test derived in Exercise 6.5.7, here are
two generated data sets. The first was generated by the R callrexp(10,1/20),
i.e., 10 observations from a Γ(1,20)-distribution. The second was generated by
rexp(12,1/40). The data are rounded and can also be found in the filegenexpd.rda.
(a)Obtain comparison boxplots of the data sets. Comment.
(b)Carry out theF-test of Exercise 6.5.7. Conclude in terms of the problem at
level 0.05.
x: 11.1 11.7 12.7 9.6 14.7 1.6 1.7 56.1 3.3 2.6
y: 55.6 40.5 32.7 25.6 70.6 1.4 51.5 12.6 16.9 63.3 5.6 66.7